Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management

Pages: 791 - 807, Volume 22, Issue 3, May 2017      doi:10.3934/dcdsb.2017039

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Renhao Cui - Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China (email)
Haomiao Li - Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, United States (email)
Linfeng Mei - Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007, China (email)
Junping Shi - Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, United States (email)

Abstract: A reaction-diffusion logistic population model with spatially nonhomogeneous harvesting is considered. It is shown that when the intrinsic growth rate is larger than the principal eigenvalue of the protection zone, then the population is always sustainable; while in the opposite case, there exists a maximum allowable catch to avoid the population extinction. The existence of steady state solutions is also studied for both cases. The existence of an optimal harvesting pattern is also shown, and theoretical results are complemented by some numerical simulations for one-dimensional domains.

Keywords:  Diffusive logistic equation, harvesting, protection zone, bifurcation.
Mathematics Subject Classification:  Primary: 35J55, 35B32, 92D25, 92D40.

Received: October 2015;      Revised: December 2015;      Available Online: January 2017.